nLab
manifold

Contents

Idea

A manifold is a topological space which looks locally like a Cartesian space, commonly a finite-dimensional Cartesian space n\mathbb{R}^n but possibly an infinite-dimensional topological vector space-

What “locally looks like” means depends on what sort of structure we are considering a Cartesian space to embody. At one extreme, we can think of n\mathbb{R}^n as merely a topological space. Or, n\mathbb{R}^n may be considered as carrying more rigid types of structure, such as C kC^k-differential structure, smooth structure, piecewise-linear (PL) structure, real analytic structure, affine structure, hyperbolic structure, foliated structure, etc., etc. Accordingly we have notions of topological manifold, differential manifold, smooth manifold, etc. By default these are modeled on finite dimensional spaces, but most notions have generalizations to a corresponding notion of infinite dimensional manifold.

In any case, the type of geometry embodied in a particular flavor of manifold is controlled by a particular groupoid of transformations which preserves whatever geometric features one is interested in; cf. Felix Klein’s Erlanger Programm.

Definitions

To give a reasonably general notion of manifold, we first specify the kinds of concrete geometric groupoids which come into play.

Pseudogroups

Definition

A pseudogroup on a topological space (or locale) XX is a groupoid GG each of whose objects is an open subset of XX, and whose morphisms are homeomorphisms between such open sets, satisfying the following conditions:

  • The objects cover XX. (Equivalently, in light of the last axiom, every open set of XX is an object of GG.)
  • If g:VWg: V \to W belongs to GG and UVU \subseteq V is an open set, then the restriction g| U:Ug(U)g|_U: U \to g(U) belongs to GG. (Equivalently, in light of the other axioms, every inclusion map id| U:UVid|_U: U \to V belongs to GG.)
  • (sheaf property) If g:UVg: U \to V is a homeomorphism and if there is a covering U αU_\alpha of UU such that the restrictions g| U α:U αg(U α)g|_{U_\alpha}: U_\alpha \to g(U_\alpha) are morphisms of GG, then gg is also morphism of GG.
Example

Commonly used choices for XX in def. 1 include

For the sake of concreteness, the reader may as well focus on the case X= nX = \mathbb{R}^n.

Charts

Let GG be a pseudogroup on XX.

Definition

A GG-chart on a topological space MM is an open subset UU of MM together with an open embedding

ϕ:UX. \phi: U \to X\,.

Two charts ϕ:UX\phi: U \to X and ψ:VX\psi: V \to X are compatible if

ψϕ 1:ϕ(UV)ψ(UV)\psi \circ \phi^{-1}: \phi(U \cap V) \to \psi(U \cap V)

belongs to GG.

Definition

A GG-atlas on a topological space MM is a family of compatible charts (ϕ α:U αX) α(\phi_\alpha: U_\alpha \to X)_\alpha, def. 2, such that (U α) α) α(U_\alpha)_\alpha)_\alpha covers MM. The (restricted) maps ϕ αβ=ϕ βϕ α 1\phi_{\alpha \beta} = \phi_\beta \circ \phi_{\alpha}^{-1} are called transition functions between the charts of the atlas.

Manifolds

Definition

A GG-manifold is a topological space equipped with a GG-atlas (definition 3).

Remark

This means that we can think of a GG-manifold as a space which is locally modeled on XX according to the geometry GG.

Remark

It is almost invariably the case in classical manifold theory that one requires some technical niceness properties on the topological space underlying a manifold.

Usually, in the definition of manifold it is understood that the underlying topological space

  1. is a Hausdorff topological space (if not one usually speaks explicitly of a non-Hausdorff manifold)

  2. is a paracompact topological space;

Often it is also assumed that the topology has a countable basis as well.

In the typical cases mentioned above for XX, this will mean that MM is metrizable. In many studies, for example in cobordism theory, one goes even further and assumes the manifolds are compact.

Remark

An atlas is not considered an essential part of the structure of a manifold: two different atlases may yield the same manifold structure. This is encoded by the following definition 5 of isomorphisms between manifolds.

Isomorphism of manifolds

Definition

An isomorphism of GG-manifolds f:MNf: M \to N (defined by chosen atlas structures, def. 3) is a homeomorphism ff such that

ϕ(Uf 1(V))ϕ 1Uf 1(V)ff(U)Vψψ(f(U)V) \phi(U \cap f^{-1}(V)) \overset{\phi^{-1}}{\to} U \cap f^{-1}(V) \overset{f}{\to} f(U) \cap V \overset{\psi}{\to} \psi(f(U) \cap V)

is in GG whenever (U,ϕ)(U, \phi) is a coordinate chart, def. 2 of xMx \in M, and (V,ψ)(V, \psi) is a coordinate chart of f(x)Nf(x) \in N.

If M 1M_1 and M 2M_2 are two GG-manifold structures on the same topological space MM, then M 1M_1 and M 2M_2 are considered equal as GG-manifolds if id:MMid: M \to M is an isomorphism from M 1M_1 to M 2M_2 (and hence also from M 2M_2 to M 1M_1).

Alternatively, atlases are ordered by inclusion, and two atlases define the same manifold structure on MM if they have a common upper bound. Equivalently, two atlases define the same manifold structure if each chart of one is compatible with each chart of the other. Or, one could extend any atlas to the (unique) maximal atlas containing it, which consists of all charts compatible with each of the charts in the original atlas, and simply identify a manifold structure with a maximal atlas.

Rafael: Can one define a manifold object in a category C as a G-manifold with G related to C? What would the relation between G and C be to obtain G-manifolds in C as manifold objects?

Toby: Yes, I think that this would make perfect sense; I think that we'd want GG to be an internal groupoid in CC. Note that defining things like ‘smooth manifold’ in CC might still be difficult, but we've reduced it to internalising Cart Sp in CC. (There's also the matter that the above definition takes a notion of space for granted, so you'd have to internalise that into CC too, but I'm not sure how important that really is, when I think about how the topology on a smooth manifold can be recovered from the smooth structure.)

Rafael: Can someone that knows more than me about this add the result of this question to this article so nobody have to ask again.

Toby: I'd rather not, since it's all ‘I think’ and ‘might be difficult’; it's better as a query box, moved to the bottom if necessary. But if Todd agrees with me, then maybe he'll add it.

Morphisms of Manifolds

Note: the following is tentative “original research”. It is prompted by the desire to extend the pseudogroup approach for defining general notions of manifold, so as to cover also an appropriate general notion of “map”. Comments, improvements, and corrections are encouraged – Todd.

I've read through it once, and it makes sense. I'll read through it again more carefully later. —Toby

We begin by defining the 2-poset (i.e., locally preordered bicategory) of regions, denoted RegReg. The objects are topological spaces (or locales if you prefer); the morphisms are partial functions with open domain, that is spans

XiUfYX \overset{i}{\leftarrow} U \overset{f}{\to} Y

where ff is continuous and ii is an open embedding. The spans are locally (that is, for fixed XX and YY) ordered by inclusion.

These local posets are not cocomplete, but they admit certain obvious joins: given a family of regional maps

(U α,f α):XY(U_\alpha, f_\alpha): X \to Y

the join α(U α,f α)\bigvee_\alpha (U_\alpha, f_\alpha) exists iff we have local compatibility:

f α| U αU β=f β| U αU βf_{\alpha}|_{U_\alpha \cap U_\beta} = f_{\beta}|_{U_\alpha \cap U_\beta}

for all α,β\alpha, \beta. Notice that composition on either side with a 11-cell preserves any local joins which exist.

Every coreflexive morphism r1 Xr \leq 1_X in RegReg splits: there is a map in RegReg,

Ext(r)idExt(r)iX,Ext(r) \overset{id}{\leftarrow} Ext(r) \overset{i}{\to} X,

whose opposite i op:XExt(r)i^op: X \to Ext(r) also belongs to RegReg (that is, ii is an open embedding), and the equations

i opi=1 Ext(r)ii op=ri^{op} \circ i = 1_{Ext(r)} \qquad i \circ i^{op} = r

hold. The object Ext(r)Ext(r) may be called the extension of rr. This splitting is a kind of comprehension principle familiar from the theory of allegories, among other things.

A cartology is a (locally full) subbicategory i:CRegi: C \hookrightarrow Reg such that

  • (Closure under open subspaces) If XOb(C)X \in Ob(C) and r1 Xr \leq 1_X in RegReg, then i:Ext(r)Xi: Ext(r) \to X and its opposite i opi^{op} are morphisms of CC.
  • (“Sheaf condition”) The inclusion i:CRegi: C \to Reg reflects and preserves local joins.

Intended examples include the case where the objects of CC are Euclidean spaces n\mathbb{R}^n, and morphisms are spans

(U,f): m n(U, f): \mathbb{R}^m \to \mathbb{R}^n

where ff is smooth.

Given a cartology CC, a morphism r=(U,f):XYr = (U, f): X \to Y in CC is pseudo-invertible if there exists s=(V,g):YXs = (V, g): Y \to X such that sr=1 Us \circ r = 1_U and rs=1 Vr \circ s = 1_V.

Lemma

In a cartology, the pseudo-invertible morphisms from an object XX to itself form a pseudogroup (as defined earlier).

The notion of a CC-manifold modeled on an object XX of CC is defined just as before, using the pseudogroup on XX implied by the previous lemma. In particular, we have CC-charts of an atlas structure on MM, which are morphisms in RegReg

XiUϕMX \overset{i}{\leftarrow} U \overset{\phi}{\to} M

satisfying the expected properties. We can thus speak of CC-manifolds (or (C,X)(C, X)-manifolds if we want to make explicit the modeling space XX).

Now, given a cartology CC, we define the category of CC-manifolds. Let MM be a (C,X)(C, X)-manifold and NN a (C,Y)(C, Y)-manifold. Then, a CC-morphism from MM to NN is a continuous map f:MNf: M \to N such that the RegReg-composite

U M V i ϕ 1 f ψ j X M N Y\array{ && U &&&& M &&&& V && \\ & i \swarrow && \searrow \phi && 1 \swarrow && \searrow f && \psi \swarrow && \searrow j & \\ X &&&& M &&&& N &&&& Y }

belongs to CC, for every pair of charts (U,ϕ):XM(U, \phi): X \to M and (V,ψ):YN(V, \psi): Y \to N.

These definitions need to be carefully checked against known examples (e.g., the categories TopTop, PLPL, and SmoothSmooth, among others).

Examples

If the term “manifold” appears without further qualification, what is usually meant is a smooth nn-manifold of some natural number dimension nn: a GG-manifold where GG is the pseudogroup of invertible C C^{\infty} maps between open sets of n\mathbb{R}^n. Replacing n\mathbb{R}^n here by a half-space {x n:x 10}\{x \in \mathbb{R}^n: x_1 \geq 0\}, one obtains the notion of smooth manifold with boundary. Or, replacing n\mathbb{R}^n here by the nn-cube I nI^n, one obtains the notion of (smooth) nn-manifold with (cubical) corners. Morphisms of manifolds are here called smooth maps, and isomorphisms are called diffeomorphisms. (In manifold theory, one usually reserves the term smooth function for smooth maps to \mathbb{R}.)

A topological nn-manifold is a manifold with respect to the pseudogroup of homeomorphisms between open sets of n\mathbb{R}^n. Any continuous function between topological manifolds is a morphisms, and any homeomorphism is an isomorphism. A piecewise-linear (PL) nn-manifold is where the pseudogroup consists of piecewise-linear homeomorphisms between such open sets; morphisms are called piecewise-linear (PL) maps.

One can go on to define, in a straighforward way, real analytic manifolds, complex analytic manifolds, elliptic manifolds, hyperbolic manifolds, and so on, using the general notion of pseudogroup.

Any space XX can always be turned into a manifold modelled on itself, using any pseudogroup GG. Simply take the inclusions of open sets as charts.

The tangent Bundle

Many species of manifolds (Riemannian manifold, pseudo-Riemannian manifold, symplectic manifold, and so on) involve extra structures defined on the tangent bundle of a smooth manifold. This is perhaps the most fundamental construction in manifold theory.

If MM is a smooth nn-manifold defined by an atlas (U α,ϕ α)(U_\alpha, \phi_\alpha), then we may define its tangent bundle TMT M by a gluing construction in TopTop, taking TMT M to be the quotient of the disjoint sum

αU α× n\sum_\alpha U_\alpha \times \mathbb{R}^n

obtained by dividing by the equivalence relation

(pU α,v)(pU β,g αβ(p)v)(p \in U_\alpha, v) \sim (p \in U_\beta, g_{\alpha\beta}(p) v)

where pU αU βp \in U_\alpha \cap U_\beta, and g αβ(p)GL( n)g_{\alpha\beta}(p) \in GL(\mathbb{R}^n) is the result of differentiating the transition function ϕ αβ\phi_{\alpha\beta} at the point ϕ α(p)\phi_\alpha(p). We thus obtain a covering U α× nU_\alpha \times \mathbb{R}^n of TMT M, and these form coordinate charts of a smooth manifold structure on TMT M in a more or less evident way. There is an obvious projection map π:TMM\pi: T M \to M, called the tangent bundle; the fiber π 1(p)\pi^{-1}(p) over a point pMp \in M is called the tangent space at pp, denoted T pMT_p M. Elements vT pMv \in T_p M are called tangent vectors at pp.

  • It is not immediately apparent that this construction yields the same manifold (in the sense described earlier) independent of the atlas chosen. To make this manifest, it is preferable to deal with coordinate-free expressions, defining for example tangent vectors with reference to the sheaf of smooth functions on MM. We discuss this below.

The functions

g αβ:U αU βGL( n)g_{\alpha \beta}: U_\alpha \cap U_\beta \to GL(\mathbb{R}^n)

satisfy Čech 1-cocycle relations

g αγ=g βγg αβ:U αU βU γGL( n)g_{\alpha \gamma} = g_{\beta\gamma} \circ g_{\alpha\beta}: U_{\alpha} \cap U_\beta \cap U_\gamma \to GL(\mathbb{R}^n)
g αα=1:U αGL( n)\qquad g_{\alpha\alpha} = 1: U_{\alpha} \to GL(\mathbb{R}^n)

These 1-cocycle data make the tangent bundle an nn-dimensional vector bundle with structure group GL( n)GL(\mathbb{R}^n).

Generalizations

References

  • John Loftin, The real definition of a smooth manifold (pdf)

Revised on July 15, 2014 00:17:05 by Urs Schreiber (89.204.138.136)